Functions, Domain, and Range

Functions

Vertical Line Test: A graph represents a function if any vertical line drawn intersects the curve at no more than one point.
- If a vertical line intersects the graph at more than one point, the graph does not represent a function.
- Example:
  - A curve y = f(x) where any vertical line intersects at only one point is a function.
  - A curve where a vertical line intersects at multiple points is not a function.

Given sets A and B, the Cartesian product A × B is a set of ordered pairs.
- Example: If A = {1, 2} and B = {2, 3}, then A \times B = {(1, 2), (1, 3), (2, 2), (2, 3)}.
Each member of the set A × B is an ordered pair (e.g., (1, 2), (1, 3)).

A function is a special type of relation in which no two distinct elements (ordered pairs) have the same first entry.
- If (a, b) is an element, a is the first entry, and b is the second entry.
In a function, if two ordered pairs have the same first entry, their second entries must also be the same.

Example A: f = {(1, 2), (-1, 2), (3, 4), (2, 6)} is a function because all first entries are distinct.
Example B: g = {(2, -1), (3, 5), (2, 4)} is NOT a function because the first entry 2 is repeated with different second entries (-1 and 4).
- For g to be a function, If the first entries are the same, the second entries must also be the same.
Example C: h = {(a, d), (c, d), (b, d)} is a function because the first entries (a, c, b) are distinct.

The domain of a function is the collection of all first entries (x-values) in the ordered pairs.
- D = {x : y = f(x)\text{ is defined}}
- Analogy: The domain represents the valid inputs into a CPU (Central Processing Unit), where the output is a real number.
- If an input results an undefined or error output, that input is not in the domain.

The range of a function is the collection of all second entries (y-values) in the ordered pairs.
- For the same example f = {(1, 2), (-1, 2), (3, 4), (2, 6)} the range is R_f = {2, 4, 6}.
  - Note: Elements in a set are not repeated.

Let y = f(x) be a function.
- The domain (D) of f is the collection of x-values such that f(x) is defined (a real number).
- The range (R) is given by the y-values (f(x)) for x in D.

Consider f(x) = \frac{1}{x - 1}.
- If x = 0, f(0) = \frac{1}{0 - 1} = -1 (real number), so 0 is in the domain.
- If x = 1, f(1) = \frac{1}{1 - 1} = \frac{1}{0} (undefined), so 1 is not in the domain.
- The domain of f(x) is all real numbers except x = 1.
In interval notation: (-\infty, 1) \cup (1, \infty).

Consider g(x) = \sqrt{x - 2}.
- If x = -1, g(-1) = \sqrt{-1 - 2} = \sqrt{-3} (not real), so -1 is not in the domain.
- If x = 0, g(0) = \sqrt{0 - 2} = \sqrt{-2} (not real), so 0 is not in the domain.
To find the domain, set the expression inside the square root to greater than or equal to zero:
- x - 2 \geq 0
- x \geq 2
- The domain of g(x) is all real numbers greater than or equal to 2.
In interval notation: [2, \infty).
Graphing the function shows that x must be at least 2, with y increasing from 0.
For a function of the form sqrt{x-a}, the function starts at the the x value of 'a' and continues to increase.

Let f be a function with domain D. Then the range R is given by the y values f(x) such that x \in D

Consider h(x) = \frac{1}{\sqrt{x - 2}}.
To find the domain, set the expression inside the square root to zero.
- \sqrt{x - 2} = 0
- x - 2 = 0
- x = 2
If x < 2, the output is not real.
There isn't a hole at 2 because the formula does not exist.
In interval notation: (2, \infty).
To find the range, find all possible output values in interval (2, \infty)
As x gets very large, the denominator also gets very large too.
Thus, the reciprocal would get very small getting closer to 0.
Thus, the range of all posible y values is in the form (0,1].